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Harmonic Conjugate Function


The harmonic conjugate to a given function u(x,y) is a function v(x,y) such that

 f(x,y)=u(x,y)+iv(x,y)

is complex differentiable (i.e., satisfies the Cauchy-Riemann equations). It is given by

 v(z)=int_(z_0)^zu_xdy-u_ydx+C,

where u_x=partialu/partialx, u_y=partialu/partialy, and C is a constant of integration.

Note that u_xdy-u_ydx is a closed form since u is harmonic, u_(xx)+v_(yy)=0. The line integral is well-defined on a simply connected domain because it is closed. However, on a domain which is not simply connected (such as the punctured disk), the harmonic conjugate may not exist.


See also

Cauchy-Riemann Equations, Complex Differentiable, Hardy Space, Harmonic Function, Hilbert Transform, Simply Connected

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Harmonic Conjugate Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HarmonicConjugateFunction.html

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